CN109328451B

CN109328451B - System and method for precoding faster-than-nyquist signaling

Info

Publication number: CN109328451B
Application number: CN201780024237.5A
Authority: CN
Inventors: 瑞莫·简娜; 吉巴克·密特拉; 鲁兹·汉斯-乔希姆·兰佩; 艾哈迈德·穆罕默德·易卜拉欣·米得拉
Original assignee: Huawei Technologies Canada Co Ltd
Current assignee: Huawei Technologies Canada Co Ltd
Priority date: 2016-04-21
Filing date: 2017-02-16
Publication date: 2021-06-04
Anticipated expiration: 2037-02-16
Also published as: CN109328451A; US10003390B2; WO2017181269A1; US20170310373A1

Abstract

A system and method for precoding faster-than-nyquist (FTN) signaling is provided. In a transmitter, thomlinson-harashima precoding (THP) is applied to produce precoded symbols. THP is based on inter-symbol interference (ISI) due to the use of super-nyquist (FTN) signaling. In the receiver, no inverse modulo operation is performed. In contrast, in the receiver, the k-th received symbol is determined by determining the nth bit b_nThe calculated log posterior probability ratio LAPPR value and the pre-calculated prior probability of the expanded constellation for a given combination of pulse shape h (t) and τ, based on the matched filter output, perform FTN processing.

Description

System and method for precoding faster-than-nyquist signaling

This application claims priority from U.S. provisional application No. 62/325,758 filed on 21/4/2016 and U.S. patent application No. 15/341,227 filed on 2/11/2016.

Technical Field

The present invention relates generally to Faster-than-Nyquist (FTN) transmission and to channel equalization and decoding techniques for high-speed digital communication systems, and more particularly to methods and apparatus for generating soft outputs for communication systems using Faster-than-Nyquist transmission.

Background

The large increase in demand for high bandwidth for network-based applications (e.g., cloud computing, video-on-demand, telepresence, etc.) has resulted in a pressing need to increase the data rate of current backbone transport networks (e.g., those provided by fiber optic links). The data rate can be increased by improving the spectral efficiency, i.e. the bandwidth (hertz) of the existing fiber link in bits per second. A potential solution to the growing spectral efficiency requirements for optical fiber communications is the use of non-orthogonal transmission through FTN signaling. FTN signaling is a linear modulation scheme that improves spectral efficiency by reducing the time and/or frequency separation between two adjacent pulses, thereby introducing inter-symbol interference (ISI) and/or inter-carrier interference (ICI). Alternatively, FTN is a technique that allows for increased bit rates while maintaining signaling bandwidth by sending faster data-carrying pulses than recommended by the "nyquist criterion" for ISI-free transmission. Achieving higher transmission rates is possible if it can be guaranteed that ISI introduced by FTN transmissions can be adequately compensated, only nominally increasing the signal-to-noise ratio (SNR) of the signal, at the expense of relatively high receiver complexity.

Furthermore, modern high performance communication systems typically employ complex forward-correction codes such as turbo codes, low-density parity-check (LDPC) codes, and the likeAn error (FEC) code to reduce the overall Bit Error Rate (BER). When FTN is used in conjunction with FEC, maximum-likelihood sequence estimation (MLSE) based on the soft Viterbi algorithm (SOVA) and Maximum A Posteriori (MAP) symbol probability methods based on the Bahl-Cocke-Jelinek-raviv (bcjr) algorithm are considered to be practical near-optimal methods of implementing FTN equalization to produce the input to the FEC decoder. See, for example, "Receivers with and without Turbo Equalization over Nyquist with and without Turbo Equalization" (Receivers for fast-than-Nyquist signalling with and without Turbo Equalization) published in IEEE int.symp.on inf.theory by a Prlja, j.b.anderson and f.rusek, a "Receivers with Reduced complexity for strong inter-symbol narrow-band Interference introduced over Nyquist signalling" (Receivers for strong inter-symbol narrow-band Interference introduced over Nyquist signalling) published in IEEE trans.commu, and j.yu, j.park, f.rusek, b.kuyadri and b.2014.201480. borscholar^thTechnique, conf, fall (VTC) "High Order Modulation in fast-through-Nyquist Signaling communications Systems" published in veh.

The computational complexity of the equalization scheme described above can be very high. In suboptimal low complexity receivers, linear equalizers and decision-feedback equalizers (DFEs) are potential candidates, but suffer from performance degradation, and in DFEs error propagation is a known major problem.

Tomlinson-Harashima Precoding (THP) has been used in systems to pre-compensate for ISI introduced by the channel. However, conventional applications of THP at the transmitter rely on feedback of channel information from the receiver to estimate the ISI introduced by the channel. For more details on the conventional application of THP, see, for example, m.tomlinson, 1971, "New Automatic Equalizer using Modulo Arithmetic (New Automatic Equalizer amplification module aritmetric)" at Electronics Letters, and h.harashima and h.miyakawa, IEEE trans.com.1972, "Matched-Transmission Technique for Channels with inter symbol Interference.

Applications of THP include modulo operations at the transmitter and receiver and are known to suffer from associated "modulo losses". Modulo operation of a THP receiver in a system that pre-compensates for ISI introduced by a channel maintains a received signal at a level for M-ary Pulse Amplitude Modulation (PAM) symbols (or equivalently, M symbols)²Quadrature Amplitude Modulation (QAM) symbols) within the modulo limit M, M), which may cause (in a low to medium signal-to-noise ratio (SNR) scheme) the received symbols to be wrapped around to the wrong side of the constellation. This in turn can cause erroneous log-likelihood ratio (LLR) calculations, which are required by FEC (e.g., LDPC) decoders as inherent information. These inaccurate LLR values can result in a significant degradation of bit-error-rate (BER) performance. For example, in coherent optical systems where the FEC input needs to reach a certain threshold BER in order to guarantee error-free transmission of LLR outputs, this condition may no longer hold due to erroneous LLR distributions caused by the modulo loss.

Disclosure of Invention

Systems and methods are provided that use FTN signaling in conjunction with THP precoding to increase bandwidth utilization. THP precoding is introduced in the transmitter to pre-compensate for ISI effects due to the use of FTN signaling. At the receiver, the LLRs are computed based on the expanded constellation, rather than performing the inversion of the modulo operation. Advantageously, the problem of mode extraction loss present in conventional THP receivers is alleviated.

The broad aspects of the invention provide a method involving applying thomlinson-haradyma precoding (THP) in a transmitter to produce precoded symbols, wherein the applying THP is based on at least one input representing inter-symbol interference (ISI) due to the use of faster-than-nyquist (FTN) signaling. Pulse shaping the precoded symbols with a faster-than-Nyquist (FTN) pulse shape. Transmitting a signal based on the pulse shaped output.

In another broad aspect, there is provided a transmitter having: a Forward Error Correction (FEC) encoder; a coherent optical transmitter comprising a Quadrature Amplitude Modulation (QAM) mapper, a Thomlinson-Harashima precoder (THP), an FTN pulse shaper, and a digital-to-analog converter; and an electro-optic front end. The THP generates precoded symbols by applying Thomson-Harashima precoding based on at least one input representing inter-symbol interference (ISI) due to the use of super-Nyquist (FTN) signaling. The pulse shaper pulse shapes the precoded symbols using the faster-than-nyquist (FTN) pulse shape.

In another broad aspect, there is provided a method comprising receiving a signal after transmission over a channel, the signal comprising symbols v [ k ] of an expanded constellation multiplied by a FTN pulse shape characterized by a roll-off factor β and a time acceleration factor τ. The method performs matched filtering on the received signal based on the FTN pulse shape to produce a matched filter output. Performing FTN processing based on the matched filter output according to a log posterior probability ratio LAPPR value calculated by determining the nth bit of the kth received symbol and a prior probability of the expanded constellation pre-calculated for a given combination of pulse shape and time acceleration factor, without performing a modulo M operation. Performing FEC decoding based on the LAPPR value.

In another broad aspect, there is provided a receiver having: a photovoltaic front end; a coherent optical receiver comprising an analog-to-digital converter (ADC), a matched filter, a polarization-mode dispersion (PMD) compensator, a log-likelihood ratio (LLR) generator from THP symbols; and a soft-decision FEC decoder. The LLR generator from the THP symbols generates soft decisions for use by the soft-decision FEC decoder using one of the methods outlined above or described herein.

Drawings

Embodiments of the present invention will be described in detail with reference to the accompanying drawings, in which:

FIG. 1A is a block diagram of a coherent optical system;

fig. 1B is a flow diagram of a method of transmitting precoded FTN signaling;

FIG. 2 is a block diagram of an FTN-THP system;

fig. 3 is an example of an extended constellation for BPSK transmission;

FIG. 4A is a schematic block diagram of a transmitter;

fig. 4B is a flow chart of a method of receiving precoded FTN signaling;

fig. 5 shows a graph of an example gain of EAD for QPSK at 0.85;

fig. 6 shows a graph of an example gain of the EAD for 16-QAM at different values of τ ═ 0.85,0.9 ];

FIGS. 7A and 7B show graphs of an example improved BER for a sliding window EAD; and

fig. 8 shows an example of autocorrelation of demapping symbols.

Detailed Description

A first embodiment of the present invention provides a simplified system and method for precoding FTN transmissions using THP, referred to herein as an FTN-THP system and method. An optical coherent transmission link consists of a transmitter (Tx), a fibre channel and a coherent receiver (Rx). A specific example of a block diagram of an FTN-THP transmitter and receiver is depicted in fig. 1A. In the transmitter, there is an FEC encoder 100 that FEC encodes data 102 from a data source. The output of the FEC encoder 100 is input to a coherent optical transmitter 104, the coherent optical transmitter 104 including a symbol mapper 106, a THP filter 108, a pulse shaping and FTN transmit processor 110, and a digital-to-analog converter (DAC) 112. The function of the THP filter 108 and the function of the pulse shaping and FTN transmit processor 110 will be described by way of example. The coherent optical transmitter is connected to a transmit opto-electronic front end 119, which transmit opto-electronic front end 119 is in turn connected to the fiber channel 116. The transmit opto-electronic front end 119 may include, for example, a linear analog driver that provides an input signal to an optical modulator, such as a Mach-Zehnder modulator (MZM) that converts an electrical analog signal to an optical signal.

In a dual-polarized system, there are two orthogonal linear polarization components (X and Y), where each component also includes two quadrature-phase components (in-phase I and quadrature Q) having the same carrier frequency. The carrier frequency is the wavelength of light provided by the laser.

The fiber channel 116 may include optical filters (e.g., cascaded Wavelength Selective Switches (WSS)), as well as optical fibers and amplifiers that are sources of Chromatic Dispersion (CD), Polarization Mode Dispersion (PMD), Polarization Dependent Loss (PDL), polarization rotation, and multiplicative phase noise.

In the receiver, there is a receive opto-electronic front end 120 connected to the fibre channel 116. The receive opto-electronic front end 120 is connected to a coherent optical receiver 130. The receive opto-electronic front end 120 may, for example, include a Polarization Beam Splitter (PBS) that separates the received optical signal into its constituent orthogonal polarizations. Typically, the PBS for a coherent optical receiver also receives a mixed signal from a local laser having the same frequency as the laser used at the transmitter. Then, a 90 degree optical hybrid is used to separate the I and Q components of each polarization, which produces four signal data paths, namely XI, XQ, YI, and YQ. Each of these constituent signals is then converted from the optical to the electrical domain by a photodetector, followed by a trans-impedance amplifier (TIA), which may provide an input to an analog-to-digital converter (ADC) 132. The digitized signal at the output of the ADC132 is provided as input to a matched filter 134, a Polarization Mode Dispersion (PMD) compensator 136, and a bit LLR generator 138 from the THP symbol, the matched filter 134 compensating for dispersion effects. The function of LLR generator 138 is described below by way of example. The soft output of the LLR generator 138 is fed to a soft decision FEC (S-FEC) decoder 140. Other aspects of signal conditioning may be performed at one or more Digital Signal Processing (DSP) modules that process the signal and recover the data. The functions following the ADC132 are typically implemented in one or more Digital Signal Processing (DSP) blocks.

Referring now to fig. 1B, shown is a flow diagram of a method of transmitting precoded faster-than-nyquist signaling provided by an embodiment of the present invention. The method may be implemented in the transmitter of fig. 1A, for example. In this specification, further details of possible implementations of the method are provided. The method begins at block 150, where thomson-harassman precoding (THP) is applied in a transmitter to produce precoded symbols, where the applied THP is based on at least one input representing inter-symbol interference (ISI) due to the use of faster-than-nyquist (FTN) pulse signaling. In block 152, the precoded symbols are pulse shaped with a faster-than-nyquist (FTN) pulse shape. In block 154, a signal is transmitted based on the pulse shaped output.

Fig. 2 depicts a simplified FTN-THP system block diagram that describes the operation of a process for THP-based FTN transmission and reception with Additive Gaussian Noise (AGN) (which may be white Gaussian noise in some embodiments) from a channel. At the transmitting end, the PAM symbols a [ k ]200 are accumulated with d [ k ]202 using an adder 203 to generate vk ]204, vk 204 being the point on the expanded constellation of the PAM under consideration. The above-mentioned increased d k corresponds to a modulo 2M operation, which will be described in more detail below with reference to FIG. 4A. The symbols v k 204 from the expanded constellation are passed through a THP feedback loop 206 with a THP filter 208 in the feedback path. The output f k 210 of the THP filter 208 is subtracted from v k. The output of the feedback loop 206 is x [ k ]212, which x [ k ]212 is input to a pulse shaper 214 that applies the FTN pulse shape h (t) to the THP precoded symbols. In a specific example, the FTN pulse shape may be a Root Raised Cosine (RRC) pulse shape. The output of the pulse shaper is transmitted through the AGN channel as depicted by the addition of the AGN component n [ k ] 216.

When FTN is used, as shown in the embodiments below, data transmission may be represented as follows:

where 0< τ ≦ 1 is the FTN time acceleration factor, τ ≦ 1 for the nyquist signal, and 0< τ <1 for FTN signaling.

At the receiver, matched filtering based on h x (-t) is performed in a matched filter 220, where x represents the complex conjugate and t represents the time reversal to generate the output of the matched filter x' [ k ] 221. The output x 'k of the matched filter is processed by an optional whitening filter 222 to produce v' k 224. In the simplified example system of FIG. 2, v '[ k ]224 is demapped by demapper 226 to generate a' [ k ] 228. It should be understood that fig. 2 only shows certain functional blocks of FTN and THP. Some other components that may be present are described by way of example in the system diagram of FIG. 1A.

In fig. 2, the FTN time acceleration factor is denoted by τ and the pulse shape h (t) is assumed to be a Root Raised Cosine (RRC) pulse with a roll-off factor characterized by β, but other pulse shapes are possible. In the particular example described, to use a single-phase filter and a minimum-phase filter for zero-forcing THP (ZF-THP), the spectral decomposition of the overall discrete-time channel, i.e., the Raised Cosine (RC) impulse response of the τ T-samples, is used to obtain the filter response of the THP filter 208.

For the current example using ZF-THP, an equivalent Forney observation model can be obtained from the Engerberg (Ungerboeck) model described above. To ensure that this is possible when RRC pulse shaping is employed, consider

FTN case (2). It will be appreciated that this is not a hard requirement for the relationship between τ and β. Different relationships between τ and β can be supported by using different spectral decompositions of the overall channel impulse response. The minimum phase component after spectral decomposition is used for the THP filter 208 and the inverse of the maximum phase component is used for the whitening filter 222 at the receiver after the matched filter 220.

The spectral decomposition satisfies:

an inherent advantage of the THP pre-filter design for ISI due to FTN transmission over conventional THP pre-filter designs for channels is that the spectral decomposition and resulting THP filter 208 does not rely on receiving any feedback from the receiver regarding the ISI introduced by the channel itself. Instead, spectral decomposition and the resulting ISI are determined by considering the known ISI due to FTN transmission in a signal s (t) that is a function of the pulse shape h (t) (e.g., parameter β for RRC pulse) and the FTN time acceleration factor τ. Compensation for channel-induced ISI is done at the receiver in accordance with conventional receiver design principles, e.g., using a linear equalizer with a limited number of taps.

Extended a priori demapper

Another embodiment provides a soft Demapper, referred to herein as an extended a-priority Demapper or "EAD" for THP systems as referred to below. In a specific example, the EAD is applied in the FTN-THP system as described above, in which case the demapper 226 is implemented as EAD. The described EAD can compensate for most of the modulo loss inherent in the demapper of a conventional THP-based transmission and outperform existing THP demappers with significant margin and can make THP competitive for optimal MAP equalization even when compared based on peak SNR.

Two methods are provided for a system based on soft-FEC decoding to calculate LLR (or log a-posterior probability ratio, LAPPR) values for bits corresponding to received symbols. This calculated LAPPR value is then provided as input to a subsequent FEC decoder 140 in a non-iterative manner. These two methods are the memoryless EAD method and the EAD sliding window method.

Memoryless EAD

Referring again to fig. 1A, consider a system that transmits Pulse Amplitude Modulation (PAM) symbols. Due to the fact thatThe effective ISI taps due to the T sampled pulses of the PAM-based FTN transmission are real-valued, so any M²The QAM constellation can be seen as an M-ary PAM modulation of I and Q. A [ k ] for PAM symbol]The input v' k of the mapper is represented and as a result of the above-mentioned model of the THP system]Are extended constellation points of AGN corruption. Fig. 3 shows an exemplary extended constellation (without AGN) in case of binary phase-shift keying (BPSK) transmission.

Conventional THP with soft detection uses a modulo operation before the demapper stage before the LAPPR value is calculated.

In accordance with one embodiment of the present invention, no modulo operation is performed in the receiver in order to reduce the modulo loss associated with conventional THP. Instead, based on the received symbol v' [ k ]]And pre-computed prior probabilities v k of the expanded constellation]Computing the nth bit b of the kth received symbol_nLAPPR value of (a), pre-computed prior probability of the expanded constellation v [ k [ ]]For a given pulse shape h (t) (e.g., parameter β for RRC pulse) and FTN parameter τ. In a specific example, the following calculation may be performed:

wherein, C_tIs a symbol set in an expanded constellation diagram set, and the nth specified bit b in the expanded constellation diagram set_nT, where t is 0 or 1. Expressing the prior probability as alpha_i＝P(v[k]＝c_i) And beta_j＝P(v[k]＝c_j) Equation (1a) can be written as

Wherein d is_i＝|v′[k]-c_i|²And σ²Is the AGN variance.

Furthermore, if

And is

Equation (1b) can be approximated as

Although equation (1b) gives an exact expression for LAPPR, equation (2) only considers an approximation of the nearest neighbor symbol of the extended constellation, which represents the nth bit equal to 1 or 0 with respect to the received symbol v' [ k ].

With an expanded constellation, as in the example of fig. 3, there is a modulo limit beyond which modulo operations occur in the transmitter (corresponding to an appropriate increase of d k in fig. 2)]202). More specifically, the signal v' k at the receiver](224 in FIG. 2) is the intermediate signal v [ k ]](204 in fig. 2), whose elements are from the expanded constellation. Due to the modulo operation at the transmitter (corresponding to the appropriate addition of d k in fig. 2)]202) Resulting in an extension of the symbol constellation beyond the modulo limit (e.g., as shown in exemplary fig. 3). In equation (2), the first term is actually an offset term to account for a given b_nAnd possible values of (c) and v [ k ]]A known relationship between probabilities outside of the constellation. Knowing τ, this probability is a prior probability used to determine the first term in equation (2).

One specific example is provided for BPSK. The following is a prior probability P (v [ k ]) required for the calculation of LLR for an analysis]＝c_i) The method of (1). The probability is derived for BPSK, but the method can be extended to any other constellation. According to the equivalent linear structure of the THP transmitter in FIG. 2, the modulus operation is formed by adding a unique sequence d [ k ] to the data symbol a]Instead, so that the precoded symbols x [ k ]]In the interval [ -2,2), it can be seen that when-4 i-2 ≦ a [ k ]]-f[k]≦ -4i +2, where i ∈ Z, where a [ k ]]Can be assumed to take on values of + -1 and

the pdf of (a) is a uniform i.i.d sequence of the output of the feedback filter, d k]4i can be well approximated by a zero mean gaussian distribution, i.e.

The probability of expanding the constellation set can be written as:

and in a similar manner to that described above,

wherein

The prior probabilities of the complete set of expanded constellations are constructed.

Fig. 4A depicts a THP transmitter structure 402 with a modulo operation 404 that is equivalent to the THP transmitter structure of fig. 2 (including adder 203 and THP feedback loop 206), where the THP transmitter structure of fig. 2 is again depicted at 400 for convenience.

In FIG. 4A, v [ k ]]＝a[k]+d[k]Wherein, for an M-ary PAM constellation, d [ k ]]2iM (I ∈ I). Can assume a k]Are independent identical distributed (i.i.d) sequences using M-PAM values. This may be ensured, for example, by using an interleaver after the FEC encoder at the transmitter. Output V of feedback filter₁(z) -1 uses

Represents, corresponds to the input x [ k ]]。f[k]Can be approximated by a zero-mean gaussian distribution, i.e., f k]～N(0,σ_f ²)。

If the set of M values of the M-ary PAM constellation is given by:

where k is 1,2, …, M, then the set V ═ { a +2iM: I ∈ I } represents the set of all odd integers that form the set of extended constellations, the set

Forming partitions of set V.

From the construction of the equivalent block diagram shown in fig. 4A, it can be seen that for any M-ary PAM constellation, when-2 iM-M ≦ a [ k ] -f [ k ≦ -2iM + M, d [ k ] ≦ 2 iM. Therefore, the temperature of the molten metal is controlled,

these values can be used to calculate the LAPPR described above. For example, such an approach may have a relatively low computational complexity compared to an all-MAP decoder.

In some implementations, for a given pulse shape (e.g., specified by the parameter β of the RRC pulse), it may yield the best performance for some τ ≧ 1/(1+ β), such that there is no expansion in the constellation diagram. For a given pulse shape and τ, the maximum value of the spread v' [ k ] of M-ary PAM may be as follows:

there is no need to buffer the received symbols because the LAPPR is calculated based only on the currently received symbols. Thus, LAPPR calculation is not delayed; the LAPPR can be calculated as long as the currently received symbol is obtained.

EAD-sliding window metric computation

Although as described aboveThe memoryless EAD method considers a one-dimensional probability density function of the symbols of the expanded constellation, but does not collect the correlation between the symbols. Fig. 8 shows an example of autocorrelation of demapping symbols. As an extension to the previous method, a second method is provided to calculate LAPPR of the nth bit corresponding to the current kth symbol based on L preceding symbols and L succeeding symbols. This approach may yield better performance than the memoryless EAD described previously. As shown in the example in fig. 8, the correlation caused by the THP filter is partially gathered in this metric, which may result in improved performance in the form of reduced BER after the FEC decoder compared to the memoryless EAD method. This method requires the pre-calculation and storage of the joint probability p (v)_k-L,v_k-L+1,…,v_k,…,v_k+L). The LAPPR value can be calculated as follows:

wherein the content of the first and second substances,

&

is a 2L +1 length spreading symbol sequence,

is a joint prior probability associated with a sequence of nth bits of 1 and 0 respectively

Similar to equations (1a) and (1b), equation (5) may be approximated as:

wherein the content of the first and second substances,

and

wherein, X₁＝(v[k-L],…,v[k]＝c_i,…v[k+L])&X₀＝(v[k-L],…,v[k]＝c_j,…v[k+L]) Is the joint probability of a 2L +1 length sequence of extended constellation symbols.

For a given window length 2L +1, sequence 2 is based on the pulse shape h (t) and τ^(2L+1)Has a zero probability. Thus, the storage requirements and number of terms appearing in the numerator and denominator are still minimal compared to the MAP and MLSE equalization methods.

The EAD metric does not depend on the whiteness of the noise, only on the noise variance. The sliding window-EAD or multidimensional EAD metric uses the white noise assumption, i.e., the noise samples in the 2L +1 length window are uncorrelated, when it is true, then the method provides the best performance. However, non-white noise does not use this metric, although there may be some performance loss. As mentioned above, there are other modules within the receiver DSP of modern communication systems that can whiten noise without explicit noise whitening filters.

Referring now to fig. 4B, shown is a flow diagram of a method of receiving precoded faster-than-nyquist signaling provided by an embodiment of the present invention. The method may be implemented in the receiver of fig. 1A, for example. In this specification, further details of possible implementations of the method are provided. The method begins at block 160 where, after transmission through a channel, a signal is received that includes an expanded constellation multiplied by a FTN pulse shape characterized by a roll-off factor β and a time acceleration factor τSymbol v [ k ] of the figure]. The method continues at block 162 where matched filtering is performed on the received signal based on the FTN pulse shape to produce a matched filter output x' [ k ]]. In block 164, without performing the modulo-M operation, FTN processing is performed based on the matched filter output by determining a log posterior probability ratio LAPPR value according to x' [ k ] and a pre-computed prior probability of the expanded constellation for a given combination of pulse shapes h (t) and τ]Nth bit b for kth received symbol_nThe calculation is performed. Based on the LAPPR value, FEC decoding is performed, block 166.

Optionally, the method further includes: whitening filtering the matched filter output x 'k to produce a whitened filter output v' k; in this case, the performing FTN processing according to x' [ k ] includes: FTN processing is performed based on the whitening-filter output v' [ k ].

In embodiments where a whitening filter is not used, the noise may be colored and whitening may be achieved in other ways, for example, a CD compensation filter in the case of optical transmission or a tapped delay line equalizer to compensate for the frequency selective radio channel.

Alternative embodiments include Feedback (FB) and Feed Forward (FF) filters, where both filters are obtained by spectral decomposition implemented at the transmitter. In this case, the transmission power is modified.

Example Performance results

To illustrate the benefits of the described method, a 1000km Standard Single Mode Fiber (SSMF) is considered, where the Chromatic Dispersion (CD) parameter value is-22.63 ps²Km, and polarization mode dispersion parameter

Other parameters considered in the simulation were: the pulse shape h (t) is an RRC pulse with β 0.3, the FEC coding rate is LDPC code rate 0.8, and the baud rate is 32/τ Gbaud/s. At the receiver, before Chromatic Dispersion (CD) and Polarization Mode Dispersion (PMD) compensationUsing a noise whitening filter, a memoryless EAD follows PMD compensation. Fig. 5 and 6 show example performance results. Fig. 7A and 7B show examples of possible improvements using a sliding window EAD, where it is shown to take into account an increase in the value of the number of symbols around the kth symbol to improve performance relative to a memoryless EAD and to reduce the gap with the optimal MAP equalizer accordingly. In fig. 7A, an example performance result of QPSK after analog forward error correction coding (post-FEC) is shown, where the performance can be seen to improve with increasing K for window lengths of 2K +1, where K is 0 (curve 1) (equivalent to no memory), and 2K +1, where K is 1,2, 3 (

curves

2, 3, 4), LDPC coding rates Rc is 0.8, β is 0.3, τ is 0.8. In fig. 7B, example performance results of simulations of QPSK are shown, where the LDPC code rate Rc is 0.8, β is 0.3, τ is 0.8 for K-0 (window length 1) (curve 1) (no memory) and for K-3 (window length 2K +1 is 7) (curve 2). FIG. 7B also shows the performance of Peh (see Peh et al, extended Soft Demapper for LDPC Coded GMD-THP MIMO System, page 519-522 of IEEE Radio and Wireless Symphosis, Peh et al, 2007) (Curve 3), MAP (best) (Curve 4), and Nyquist (ISI-free) AWGN (Curve 5). The performance of K-3 is closer to that of the optimal MAP receiver than that of K-0.

Although the present invention has been described with reference to specific features and embodiments, various modifications and combinations of these features and embodiments are possible without departing from the present invention. Accordingly, the specification and figures are to be regarded as illustrative only of some embodiments of the invention, which are defined by the appended claims, and the invention covers any and all modifications, variations, or equivalents that fall within the scope of the invention. Thus, although the present disclosure and its advantages have been described in detail, various changes, substitutions and alterations can be made herein without departing from the invention as defined by the appended claims. Moreover, the scope of the present application is not intended to be limited to the particular embodiments of the process, machine, manufacture, composition of matter, means, methods and steps described in the specification. As one of ordinary skill in the art will readily appreciate from the disclosure of the present invention, processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed that perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein may be utilized according to the present invention. Accordingly, the appended claims are intended to include within their scope such processes, machines, manufacture, compositions of matter, means, methods, or steps.

Further, any module, component, or device executing instructions exemplified herein can include or otherwise have access to non-transitory computer/processor readable storage media for storing information, such as computer/processor readable instructions, data structures, program modules, and/or other data. A non-exhaustive list of non-transitory computer/processor readable storage media includes magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, optical disks (e.g., compact disk read-only memory (CD-ROM), digital video disks or Digital Versatile Disks (DVD), blu-ray disks or other optical storage), volatile and non-volatile and removable and non-removable media implemented in any method or technology, random-access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory, or other storage technologies. Any non-transitory computer/processor storage medium may be part of or accessible or connectable to a device. Any of the applications or modules described herein may be implemented using computer/processor readable/executable instructions that may be stored or otherwise maintained by such non-transitory computer/processor readable storage media.

Claims

1. A method, comprising:

after transmission over a channel, receiving a signal comprising symbols vk multiplied by an extended constellation of a faster-than-nyquist FTN pulse shape characterized by a roll-off factor β and a time-acceleration factor τ;

performing matched filtering on the received signal based on the FTN pulse shape to produce a matched filter output x' [ k ];

does not perform the operation of modulus M, by determining the value according to x' [ k ]]Nth bit b for kth received symbol_nPerforming FTN processing based on the matched filter output, the calculated log posterior probability ratio LAPPR value and a pre-calculated prior probability of the expanded constellation for a given combination of pulse shapes h (t) and τ;

performing forward error correction, FEC, decoding based on the LAPPR value.

2. The method of claim 1, further comprising:

whitening filtering the matched filter output x 'k to produce a whitened filter output v' k;

wherein the performing FTN processing based on x' [ k ] comprises: FTN processing is performed based on the whitening-filter output v' [ k ].

3. The method of claim 2, wherein the whitening filtering has a frequency response that is a maximum phase representation of a spectral decomposition of h (t) x h (-t), wherein h (t) is a super-nyquist pulse shape.

4. The method of claim 2, wherein the determining the LAPPR value comprises employing

Or

Or

Wherein, C_tIs a symbol set in an extended constellation set, wherein the n-th designated bit b in the extended constellation set_nT is 0 or 1.

5. The method of claim 1 or 2, further comprising:

determining the LAPPR value based on the current k-th symbol and L preceding symbols and L following symbols, wherein L is more than or equal to 1.

6. The method of claim 2, wherein the determining the LAPPR value comprises using

Or

Wherein, C_tIs a set of symbols in a set of extended constellations, the nth specified bit b in the set of extended constellations_nT is 0 or 1.

7. The method of claim 1, further comprising:

applying Thomlinson-Harashima precoding (THP) in a transmitter to produce precoded symbols, wherein the applied THP is based on at least one input representing inter-symbol interference (ISI) due to using super-Nyquist FTN signaling;

applying pulse shaping to the precoded symbols using the FTN pulse shape;

transmitting a signal based on the pulse shaped output.

8. The method of claim 7, wherein applying THP in the transmitter comprises: a THP filter is used in a feedback loop within the transmitter.

9. The method of claim 8, wherein the THP filter has a frequency response that is a minimum phase representation of a spectral decomposition of h (t) x h (-t), wherein h (t) is the super-nyquist pulse shape.

10. The method of claim 8, wherein M-ary pulse amplitude modulation, PAM, signaling is employed, the method further comprising:

performing a modulo 2M operation in a forward path of the feedback loop.

11. The method of claim 8, wherein M-ary PAM signaling is employed, the method further comprising:

adding a quantity before the feedback loop, which is equivalent to performing a modulo 2M operation in the forward path of the feedback loop.

12. The method of claim 7 or 8, wherein the FTN pulse shape is a root raised cosine RRC pulse shape characterized by β and a FTN time acceleration factor τ, 0< τ < 1.

13. A receiver, comprising:

a photovoltaic front end;

the coherent optical receiver comprises an analog-to-digital converter (ADC), a matched filter, a Polarization Mode Dispersion (PMD) compensator and a log-likelihood ratio (LLR) generator from a ThmLinson-Harashima precoder THP symbol;

a soft-decision FEC decoder;

wherein the LLR generator from THP symbols generates soft decisions for use by the soft-decision FEC decoder using the method of claim 1.

14. The receiver of claim 13, wherein the LLR generator comprises a whitening filter that performs whitening filtering and has a frequency response that is a maximum phase representation of a spectral decomposition of h (t) x h (-t), wherein h (t) is a super-nyquist pulse shape.

15. The receiver of claim 13 or 14, wherein the LLR generator is configured to determine the LAPPR value based on the current kth symbol and L preceding symbols and L succeeding symbols, where L ≧ 1.

16. An optical system, comprising:

a receiver as claimed in any one of claims 13 to 15;

a transmitter, the transmitter comprising:

a forward error correction, FEC, encoder;

a coherent optical transmitter comprising a Quadrature Amplitude Modulation (QAM) mapper, a Thmlinson-Harashima precoder (THP), a faster-than-Nyquist (FTN) pulse shaper, and a digital-to-analog converter;

an electro-optic front end;

wherein the THP generates precoded symbols by applying Tomlinson-Harashima precoding based on at least one input representing inter-symbol interference (ISI) due to use of a super-Nyquist pulse shape; and is

Wherein the FTN pulse shaper applies pulse shaping to the precoded symbols utilizing the faster-than-Nyquist FTN pulse shape.

17. The optical system of claim 16, wherein the THP comprises: a THP filter in a feedback loop within the transmitter.

18. The optical system of claim 17, wherein the THP filter has a frequency response that is a minimum phase representation of a spectral decomposition of h (t) x h (-t), wherein h (t) is the super-nyquist pulse shape.

19. The optical system according to claim 17 or 18, wherein M-ary pulse amplitude modulation, PAM, signalling is employed, the transmitter further comprising:

a modulo 2M operator in a forward path of the feedback loop.

20. The optical system of claim 17 or 18, wherein M-ary PAM signaling is employed, the transmitter further comprising:

an adder, added by an amount before the feedback loop, corresponding to performing a modulo 2M operation in the forward path of the feedback loop.

21. The optical system according to any one of claims 16 to 18, wherein the FTN pulse shape is a root raised cosine RRC pulse shape characterized by a roll-off factor β and a FTN time acceleration factor τ, 0< τ < 1.